Linear Response Theory for Shear Modulus $C_{66} $ and Raman Quadrupole Susceptibility: Evidence for Orbital Fluctuations in Fe-Based Superconductors

Existence of strong nematic fluctuations in various Fe-based superconductors has been discussed as a central issue. To clarify the origin, we discuss both the softening of shear modulus $C_{66} $ and the enhancement of the charge quadrupole susceptibility observed by Raman spectroscopy $\chi _{x^{2}-y^{2}}^{\mbox{Raman}} $. Due to the Aslamazov-Larkin vertex correction (AL-VC), strong orbital nematic fluctuations are induced by spin fluctuations. The strong development of $1/C_{66} $ is given by the summation of the Pauli and Van-Vleck orbital susceptibilities due to AL-VC, whereas moderate enhancement of $\chi_{x^{2}-y^{2}}^{\mbox{Raman}} $ is induced by the Van-Vleck term only. Therefore, a consistent explanation for the difference behavior between two measurements is achieved based on the orbital fluctuation theory.